Lemma 4.1  [[1]} Lemma 2].
Let us define \(\Omega =\mathbb {R}^{d}\backslash \mathbb {B}(0,R)\)  where
\(\mathbb {B}(0,R)\)  is the open ball of radius \(R\)  centered at 0,
and define \(V\left(x\right)=\inf \left\lbrace  \sum _{i}\lambda _{i}U(\ x_{i})\right\rbrace  \) 
where the infimum is running over all possible convex combination
of points \(x_{i}\)  (that is \(\lambda _{i}\ge 0\) , \(\sum _{i}\lambda _{i}=1\) 
and \(\sum _{i}\lambda _{i}x_{i}=x\) ). Then for \(\forall \ x\in \mathbb {B}(0,R)\) ,
\(V(\ x)\)  can be represented as a convex combination of \(U\left(x_{j}\right)\) 
such that \(\left\Vert x_{j}\right\Vert =R,\)  that is \(V\left(x\right)=\inf \left\lbrace  \sum _{j}\lambda _{j}U(\ x_{j})\right\rbrace  \) 
where \(\lambda _{j}\ge 0\) , \(\sum _{j}\lambda _{j}=1\)  and \(\sum _{j}\lambda _{j}x_{j}=x\) 
and \(\left\Vert x_{j}\right\Vert =R.\)  Then, \(\inf _{\left\Vert \bar{x}\right\Vert =R}U(\bar{x})\le V(\ x)\le \sup _{\left\Vert \bar{x}\right\Vert =R}U(\bar{x}).\)

